Understanding the Linearity Test for Inner Products

[JamesGoh]

In one of my tutorial problems, I was asked to verify if the following function
is a valid inner product

<$x,y$>= $x1x2 + y1y2$

Note, x=(x1,x2)$^{T}$ and y=(y1,y2)$^{T}$

where T means transpose of the matrix

The tutor said to us the answer is no because it fails the linearity test

Does it fail the linearity test because of the x1 and y1 terms in front of the x2 and y2 ?

 

[antibrane]

Try explicitly calculating
$$\langle x+y,z\rangle$$
and see if it really does equal
$$\langle x,z\rangle +\langle y,z\rangle$$
If it doesn't then you know it does not satisfy linearity.

 

[zhentil]

Does <x,0*y>=0*<x,y>?

 

[HallsofIvy]

Yes, it does. But that doesn't prove anything.

 

[Landau]

Yes, it does.

No it doesn't. <x,0>=x1x2.

 

[HallsofIvy]

Oh, Blast! I was interpreting the given inner product wrong!